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Mirrors > Home > MPE Home > Th. List > midcgr | Structured version Visualization version GIF version |
Description: Congruence of midpoint. (Contributed by Thierry Arnoux, 7-Dec-2019.) |
Ref | Expression |
---|---|
ismid.p | ⊢ 𝑃 = (Base‘𝐺) |
ismid.d | ⊢ − = (dist‘𝐺) |
ismid.i | ⊢ 𝐼 = (Itv‘𝐺) |
ismid.g | ⊢ (𝜑 → 𝐺 ∈ TarskiG) |
ismid.1 | ⊢ (𝜑 → 𝐺DimTarskiG≥2) |
midcl.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑃) |
midcl.2 | ⊢ (𝜑 → 𝐵 ∈ 𝑃) |
midcgr.1 | ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = 𝐶) |
Ref | Expression |
---|---|
midcgr | ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | midcgr.1 | . . . 4 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) = 𝐶) | |
2 | ismid.p | . . . . 5 ⊢ 𝑃 = (Base‘𝐺) | |
3 | ismid.d | . . . . 5 ⊢ − = (dist‘𝐺) | |
4 | ismid.i | . . . . 5 ⊢ 𝐼 = (Itv‘𝐺) | |
5 | ismid.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ TarskiG) | |
6 | ismid.1 | . . . . 5 ⊢ (𝜑 → 𝐺DimTarskiG≥2) | |
7 | midcl.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ 𝑃) | |
8 | midcl.2 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ 𝑃) | |
9 | eqid 2725 | . . . . 5 ⊢ (pInvG‘𝐺) = (pInvG‘𝐺) | |
10 | 2, 3, 4, 5, 6, 7, 8 | midcl 28653 | . . . . . 6 ⊢ (𝜑 → (𝐴(midG‘𝐺)𝐵) ∈ 𝑃) |
11 | 1, 10 | eqeltrrd 2826 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ 𝑃) |
12 | 2, 3, 4, 5, 6, 7, 8, 9, 11 | ismidb 28654 | . . . 4 ⊢ (𝜑 → (𝐵 = (((pInvG‘𝐺)‘𝐶)‘𝐴) ↔ (𝐴(midG‘𝐺)𝐵) = 𝐶)) |
13 | 1, 12 | mpbird 256 | . . 3 ⊢ (𝜑 → 𝐵 = (((pInvG‘𝐺)‘𝐶)‘𝐴)) |
14 | 13 | oveq2d 7435 | . 2 ⊢ (𝜑 → (𝐶 − 𝐵) = (𝐶 − (((pInvG‘𝐺)‘𝐶)‘𝐴))) |
15 | eqid 2725 | . . 3 ⊢ (LineG‘𝐺) = (LineG‘𝐺) | |
16 | eqid 2725 | . . 3 ⊢ ((pInvG‘𝐺)‘𝐶) = ((pInvG‘𝐺)‘𝐶) | |
17 | 2, 3, 4, 15, 9, 5, 11, 16, 7 | mircgr 28533 | . 2 ⊢ (𝜑 → (𝐶 − (((pInvG‘𝐺)‘𝐶)‘𝐴)) = (𝐶 − 𝐴)) |
18 | 14, 17 | eqtr2d 2766 | 1 ⊢ (𝜑 → (𝐶 − 𝐴) = (𝐶 − 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 (class class class)co 7419 2c2 12300 Basecbs 17183 distcds 17245 TarskiGcstrkg 28303 DimTarskiG≥cstrkgld 28307 Itvcitv 28309 LineGclng 28310 pInvGcmir 28528 midGcmid 28648 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-oadd 8491 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-dju 9926 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-xnn0 12578 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-hash 14326 df-word 14501 df-concat 14557 df-s1 14582 df-s2 14835 df-s3 14836 df-trkgc 28324 df-trkgb 28325 df-trkgcb 28326 df-trkgld 28328 df-trkg 28329 df-cgrg 28387 df-leg 28459 df-mir 28529 df-rag 28570 df-perpg 28572 df-mid 28650 |
This theorem is referenced by: midcom 28658 lmiisolem 28672 hypcgrlem1 28675 hypcgrlem2 28676 |
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